We emphasize that \(C^0\)  conjugacy in general does not preserve important dynamical properties
such as characteristic directions and smoothness of invariant manifolds. Preserving these properties needs at least \(C^1\)  conjugacy. In 1970s, an effort was made by Belitskii in [1]} to establish a \(C^1\)  linearization
of \(C^2\)  hyperbolic diffeomorphisms on \(\mathbb {R}^d\)  under certain second order non-resonance conditions. This result requires much weaker conditions on smoothness and non-resonance than the Sternberg's theorem in \(C^1\)  case. Moreover, Hartman's example given in [2]} shows that Belitskii's non-resonance conditions cannot be avoided.
